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受欢迎的微积分 >

integral of (sqrt(7-x^4))/(5x)

解答

\int \frac{7 - x ^{4}}{5 x} d x

解答

- \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7) + C

求解步骤

\int \frac{7 - x ^{4}}{5 x} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int \frac{7 - x ^{4}}{x} d x

使用换元积分法

= \frac{1}{5} \cdot \int \frac{7 - u}{4 u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \frac{1}{4} \cdot \int \frac{7 - u}{u} d u

使用换元积分法

= \frac{1}{5} \cdot \frac{1}{4} \cdot \int - \frac{2 v ^{2}}{- v ^{2} + 7} d v

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \frac{1}{4} (- 2 \cdot \int \frac{v ^{2}}{- v ^{2} + 7} d v)

使用换元积分法

= \frac{1}{5} \cdot \frac{1}{4} (- 2 \cdot \int \frac{7 w ^{2}}{- w ^{2} + 1} d w)

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \frac{1}{4} (- 27 \cdot \int \frac{w ^{2}}{- w ^{2} + 1} d w)

\frac{w ^{2}}{- w ^{2} + 1} = \frac{1}{- w ^{2} + 1} - 1

= \frac{1}{5} \cdot \frac{1}{4} (- 27 \cdot \int \frac{1}{- w ^{2} + 1} - 1 d w)

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{5} \cdot \frac{1}{4} (- 27 (\int \frac{1}{- w ^{2} + 1} d w - \int 1 d w))

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

\int 1 d w = w

= \frac{1}{5} \cdot \frac{1}{4} (- 27 (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2} - w))

代回

= \frac{1}{5} \cdot \frac{1}{4} - 27 \frac{ln \frac{7 - x ^{4}}{7} + 1}{2} - \frac{ln \frac{7 - x ^{4}}{7} - 1}{2} - \frac{7 - x ^{4}}{7}

化简

\frac{1}{5} \cdot \frac{1}{4} - 27 \frac{ln \frac{7 - x ^{4}}{7} + 1}{2} - \frac{ln \frac{7 - x ^{4}}{7} - 1}{2} - \frac{7 - x ^{4}}{7} : - \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7)

= - \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7)

解答补常数

= - \frac{1}{20} (7 ln \frac{- x ^{4} + 7}{7} + 1 - 7 ln \frac{- x ^{4} + 7}{7} - 1 - 2 - x^{4} + 7) + C

作图

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